Atomic nuclei and the stability of isotopes

The potassium in bananas contains a small fraction of the isotope potassium-40, whose nucleus has atomic number $Z = 19$ and mass number $A = 40$.

1. a) State how many protons and how many neutrons this nucleus has.
2. b) Write its symbol using ${}^{A}_{Z}\text{X}$ notation.
3. c) How many electrons does a neutral atom of potassium-40 have? Justify your answer.

Carbon has three natural isotopes: carbon-12, carbon-13 and carbon-14, all with $Z = 6$.

1. a) Explain what it means for two nuclei to be isotopes.
2. b) Calculate the number of neutrons in the nucleus of each of the three isotopes.
3. c) Do all three isotopes have the same chemical properties? Reason your answer considering their electrons.

Consider these five nuclides: ${}^{14}_{6}\text{C}$, ${}^{14}_{7}\text{N}$, ${}^{12}_{6}\text{C}$, ${}^{16}_{8}\text{O}$ and ${}^{15}_{7}\text{N}$.

1. a) Calculate the number of protons and neutrons in each one.
2. b) State which pairs are isotopes (same $Z$) and which pair is isobars (same $A$).
3. c) Identify two nuclides that have the same number of neutrons (isotones).

In a helium nucleus, two protons are confined in a region about $10^{-15}\ \text{m}$ in diameter. As both have positive charge, they repel each other electrically with considerable force; however, the nucleus does not break apart.

1. a) Explain what the strong nuclear force is and why it manages to keep the nucleus bound together.
2. b) Reason why, even with that force existing, nuclei with too many protons end up being unstable.

Two protons inside a nucleus are separated by a distance of $10^{-15}\ \text{m}$, as shown in the figure. Data: Coulomb constant $K = 9 \cdot 10^{9}\ \text{N}\cdot\text{m}^2/\text{C}^2$ and proton charge $e = 1,6 \cdot 10^{-19}\ \text{C}$.

1. a) Calculate the repulsive electrostatic force between them using Coulomb's law, $F = K \cdot \frac{e^2}{r^2}$.
2. b) The value obtained is huge for a force between particles: explain what conclusion can be drawn about the strength of the strong nuclear force that holds them together.

The $\frac{N}{Z}$ ratio of stable nuclei is approximately $1$ for light nuclei and increases to about $1,5$ for the heaviest ones, as shown in the figure.

Calculate the $\frac{N}{Z}$ ratio for each of these nuclides and predict, by comparing with this trend, whether they are expected to be stable or unstable:

1. a) ${}^{12}_{6}\text{C}$
2. b) ${}^{14}_{6}\text{C}$
3. c) ${}^{206}_{82}\text{Pb}$
4. d) ${}^{238}_{92}\text{U}$

Precision measurements show that the mass of a helium-4 nucleus is less than the sum of the masses of its two protons and two neutrons measured separately.

1. a) Explain what the mass defect of a nucleus is.
2. b) What has the "missing" mass been transformed into? Mention Einstein's relation $E = \Delta m \cdot c^2$ in your explanation.

Data: mass of proton $m_p = 1,0073\ \text{u}$, mass of neutron $m_n = 1,0087\ \text{u}$, mass of helium-4 nucleus $m_{\alpha} = 4,0015\ \text{u}$ and $1\ \text{u} = 1,66 \cdot 10^{-27}\ \text{kg}$.

1. a) Calculate the sum of the masses of the two protons and two neutrons that form the helium-4 nucleus, measured separately.
2. b) Calculate the mass defect $\Delta m$ of the nucleus, in u units.
3. c) Express this mass defect in kilograms, in standard form.

The mass defect of the helium-4 nucleus is $\Delta m = 0,0305\ \text{u}$, equivalent to $5,06 \cdot 10^{-29}\ \text{kg}$. Data: $c = 3 \cdot 10^{8}\ \text{m/s}$ and $1\ \text{eV} = 1,6 \cdot 10^{-19}\ \text{J}$.

1. a) Calculate the binding energy of the nucleus using Einstein's relation, $E = \Delta m \cdot c^2$, expressed in joules.
2. b) Convert the result to $\text{MeV}$ (note that $1\ \text{MeV} = 10^{6}\ \text{eV}$).
3. c) Calculate the binding energy per nucleon of helium-4.

In nuclide tables, the binding energy per nucleon of iron-56 is given as $8,8\ \text{MeV}$ and that of uranium-235 as $7,6\ \text{MeV}$, as shown in the figure.

1. a) Calculate the total binding energy of each of these two nuclei.
2. b) Explain why iron-56 is more stable than uranium-235 even though its total binding energy is smaller.
3. c) Relate the result to the fact that uranium-235 releases energy when undergoing fission.

Data: mass of proton $m_p = 1,0073\ \text{u}$, mass of neutron $m_n = 1,0087\ \text{u}$, mass of iron-56 nucleus $m(\text{Fe}) = 55,9207\ \text{u}$ and $1\ \text{u} \cdot c^2 = 931,5\ \text{MeV}$.

1. a) Iron-56 has $Z = 26$: state how many protons and how many neutrons its nucleus contains.
2. b) Calculate its mass defect, in u units.
3. c) Calculate its total binding energy and its binding energy per nucleon, in $\text{MeV}$.
4. d) Given that the binding energy per nucleon of carbon-12 is $7,7\ \text{MeV}$, reason which of the two nuclei is more stable and which process, fusion or fission, could release energy starting from each one.

The curve of binding energy per nucleon against mass number $A$ increases rapidly from the lightest nuclei, reaches a maximum around iron-56 ($8,8\ \text{MeV}$ per nucleon) and then decreases gently to about $7,6\ \text{MeV}$ in uranium-235.

1. a) Sketch this curve in your exercise book, marking the maximum and the regions of light and heavy nuclei.
2. b) Explain, from the shape of the curve, why the fusion of light nuclei releases energy.
3. c) Explain, using the same argument, why the fission of very heavy nuclei also releases energy.
4. d) Reason why energy cannot be obtained by either fusing or fissioning iron nuclei.

You can do it on paper and send a photo with your answer.